Average Error: 58.5 → 0.2
Time: 29.4s
Precision: 64
\[\log \left(\frac{1.0 - \varepsilon}{1.0 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{\varepsilon}{1.0} \cdot \left(\frac{\varepsilon}{1.0} \cdot \frac{\varepsilon}{1.0}\right), \frac{-2}{3}, \frac{{\varepsilon}^{5}}{{1.0}^{5}} \cdot \frac{-2}{5}\right) - 2.0 \cdot \varepsilon\]
\log \left(\frac{1.0 - \varepsilon}{1.0 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{\varepsilon}{1.0} \cdot \left(\frac{\varepsilon}{1.0} \cdot \frac{\varepsilon}{1.0}\right), \frac{-2}{3}, \frac{{\varepsilon}^{5}}{{1.0}^{5}} \cdot \frac{-2}{5}\right) - 2.0 \cdot \varepsilon
double f(double eps) {
        double r4901673 = 1.0;
        double r4901674 = eps;
        double r4901675 = r4901673 - r4901674;
        double r4901676 = r4901673 + r4901674;
        double r4901677 = r4901675 / r4901676;
        double r4901678 = log(r4901677);
        return r4901678;
}
↓
double f(double eps) {
        double r4901679 = eps;
        double r4901680 = 1.0;
        double r4901681 = r4901679 / r4901680;
        double r4901682 = r4901681 * r4901681;
        double r4901683 = r4901681 * r4901682;
        double r4901684 = -0.6666666666666666;
        double r4901685 = 5.0;
        double r4901686 = pow(r4901679, r4901685);
        double r4901687 = pow(r4901680, r4901685);
        double r4901688 = r4901686 / r4901687;
        double r4901689 = -0.4;
        double r4901690 = r4901688 * r4901689;
        double r4901691 = fma(r4901683, r4901684, r4901690);
        double r4901692 = 2.0;
        double r4901693 = r4901692 * r4901679;
        double r4901694 = r4901691 - r4901693;
        return r4901694;
}

Error

Bits error versus eps

Target

Original58.5
Target0.2
Herbie0.2
\[-2.0 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3.0}}{3.0}\right) + \frac{{\varepsilon}^{5.0}}{5.0}\right)\]

Derivation

  1. Initial program 58.5

    \[\log \left(\frac{1.0 - \varepsilon}{1.0 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div58.5

    \[\leadsto \color{blue}{\log \left(1.0 - \varepsilon\right) - \log \left(1.0 + \varepsilon\right)}\]

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(2.0 \cdot \varepsilon + \left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1.0}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1.0}^{5}}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\varepsilon}{1.0} \cdot \frac{\varepsilon}{1.0}\right) \cdot \frac{\varepsilon}{1.0}, \frac{-2}{3}, \frac{{\varepsilon}^{5}}{{1.0}^{5}} \cdot \frac{-2}{5}\right) - 2.0 \cdot \varepsilon}\]

  6. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{1.0} \cdot \left(\frac{\varepsilon}{1.0} \cdot \frac{\varepsilon}{1.0}\right), \frac{-2}{3}, \frac{{\varepsilon}^{5}}{{1.0}^{5}} \cdot \frac{-2}{5}\right) - 2.0 \cdot \varepsilon\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))